Consider an interval $I = [t_0,t_1]$ and a finite dimensional Banach space $X$. Let $U$ be an open subset of $\mathbb{R} \times X \times X$ and let $V \subseteq \mathcal{C}^{1}(I,X)$ be the set of all curves $c:I \rightarrow X$ where $(t,c(t),c^\prime(t))$ is contained in $U$ for all $t$ equipped with the norm
$$\lvert\lvert c \rvert\rvert_{\mathcal{C}^1(I)} := \sup_{t \in I} \lvert\lvert c(t) \rvert\rvert_X + \sup_{t \in I}\lvert\lvert c^\prime(t) \rvert\rvert_X.$$
Show that $V$ is open in $\mathcal{C}^{1}(I,X)$.
My idea would be to define $\widetilde{c}$ as $\widetilde{c}(t) := c(t)+\varepsilon$, then we have
$$\lvert\lvert c-\widetilde{c} \rvert\rvert_{\mathcal{C}^1(I)} = sup_{t \in I} \lvert\lvert (c-\widetilde{c})(t)\rvert\rvert_X = \lvert\lvert \varepsilon \rvert\rvert_X = \varepsilon.$$
Is this correct?